What is private banking?

Private banking is banking, investment and other financial services provided by banks to private individuals who invest sizable assets.
The term "private" refers to customer service rendered on a more personal basis than in mass-market retail banking, usually via dedicated bank advisers.
Historically, private banking has been viewed as a very exclusive niche that only caters to High-net-worth individuals (HNWIs) with liquidity over $2 million, though it is now possible to open private banking accounts with as little as $250,000 for private investors.
An institution's private banking division provides services such as wealth management, savings, inheritance, and tax planning for their clients.
A high-level form of private banking (for the especially affluent) is often referred to as wealth management.
For private banking services clients pay either based on the number of transactions, the annual portfolio performance or a "flat-fee", usually calculated as a yearly percentage of the total investment amount.

Very well explained Frank. 

Just to elaborate further, people do open private banking accounts to maintain the high level of secrecy. All high profile and ultra high net worth individuals (UHNWI) prefers to have private banking accounts to invest their excess funds in the International financial markets such as in Singapore, Jersey, Switzerland and British Virgin island (BVI)